Question

1. the area of a rectangle is 6x² - 17x - 45 and the width is 2x - 9. what is the length?

Explanation:

Step1: Recall area formula

The area formula of a rectangle is $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width. We need to find $l=\frac{A}{w}$. Given $A = 6x^{2}-17x - 45$ and $w=2x - 9$.

Step2: Factor the area expression

Factor $6x^{2}-17x - 45$. We need to find two numbers that multiply to $6\times(- 45)=-270$ and add up to -17. The numbers are -27 and 10. So, $6x^{2}-17x - 45=6x^{2}-27x+10x - 45=3x(2x - 9)+5(2x - 9)=(3x + 5)(2x - 9)$.

Step3: Calculate the length

Since $l=\frac{A}{w}$, substituting $A=(3x + 5)(2x - 9)$ and $w = 2x - 9$, we get $l=\frac{(3x + 5)(2x - 9)}{2x - 9}=3x+5$ (assuming $x
eq\frac{9}{2}$).

Answer:

$3x + 5$